Finite Group Databasebeta

← #6510783

Subgroups — #6510783

Order 512.

2358 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3189C211yesmaximal, characteristic, complemented
2C2×#3187C211yesmaximal, characteristic
3C22×#483C211yesmaximal, characteristic, complemented
4C22×#480C211yesmaximal, characteristic
5#35019C211yesmaximal, characteristic
6C2×#2503C211yesmaximal, characteristic, complemented
7#28335C214yesmaximal, complemented
8#28330C214yesmaximal, complemented
9#28306C214yesmaximal, complemented
10#27931C214yesmaximal, complemented
11#27926C214yesmaximal, complemented
12#27912C214yesmaximal, complemented
13#20523C211yesmaximal, characteristic, complemented
14C22×C22⋊Q8C2211yescharacteristic
15C22×C22⋊Q8C2211yescharacteristic
16C22×#121C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18#3189—28no—
19#3187—28no—
20C2×#484C2212yescomplemented
21C2×#484C2212yescomplemented
22C2×#481C2212yes—
23C2×#483—28no—
24C2×#483C2212yes—
25C2×#483C2214yes—
26C2×#483C2214yescomplemented
27C2×#480—28no—
28C2×#480C2214yes—
29C2×#480C2214yes—
30C2×#467C2214yes—
31C2×#465C2214yes—
32C2×#451C2212yescomplemented
33C2×#451C2212yescomplemented
34C2×#448C2212yes—
35C2×#450C2212yes—
36C22×#107C2212yes—
37C22×#106C2212yescomplemented
38C22×#103C2212yescomplemented
39C22×#102C2212yes—
40C22×C22⋊C8C2211yescharacteristic
41C22×C22⋊C8C2211yescharacteristic
42#2850C2214yes—
43#2639—22no—
44#2591—22no—
45#2561—22no—
46#2504C2212yes—
47#2503—22no—
48#2503—22no—
49#2503C2212yes—
50#2503C2214yes—
51#2503C2214yescomplemented
52#2436C2212yes—
53#2435C2212yes—
54C2×#400—22no—
55#2308—22no—
56C2×#391C2212yescomplemented
57C2×#382C2212yescomplemented
58C2×#381—22no—
59C2×C4×C22⋊C4C2211yescharacteristic
60#2097C2218yescomplemented
61#2097C2218yescomplemented
62#2092C2218yescomplemented
63#2015C2218yescomplemented
64#1969C2218yescomplemented
65#1968C2218yescomplemented
66#1958C2218yescomplemented
67#1846C2218yescomplemented
68C8×C22⋊C4C2214yes—
69C8×C22⋊C4C2214yescomplemented
70C4×C22⋊C8C2214yescomplemented
71C23×Q8C2311yescharacteristic
72C23×Q8C2311yescharacteristic
73C23×D4—21no—
74C23×D4C2311yescharacteristic
75C24×C4—21no—
76C24×C4C2311yescharacteristic
77C22×Q16C2314yes—
78C22×SD16C2314yes—
79C23×C8C2312yes—
80C23×C8C2312yes—
81C2×#124C2314yes—
82C2×C22⋊Q8—21no—
83C2×C22⋊Q8—21no—
84C2×C22⋊Q8—21no—
85C2×C22⋊Q8—21no—
86C2×C22⋊Q8—24no—
87C2×C22⋊Q8—24no—
88C2×C22⋊Q8—24no—
89C2×C22⋊Q8—24no—
90C2×C22⋊Q8C2312yes—
91C2×C22⋊Q8C2312yes—
92C2×C22⋊Q8C2312yes—
93C2×C22⋊Q8C2312yes—
94C2×#121—21no—
95C2×#121—21no—
96C2×#121—24no—
97C2×#121—24no—
98C2×#121C2312yes—
99C2×#121C2312yes—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4D412yes—
117C22×C42C2312yes—
118#484—28no—
119#484—28no—
120#483—216no—
121#483—216no—
122#483—28no—
123#481—28no—
124#480—216no—
125#480—216no—
126#467—216no—
127#465—216no—
128#451—28no—
129#451—28no—
130#450—28no—
131#448—28no—
132C2×#107—28no—
133C2×#107C2318yes—
134C2×#107C2314yes—
135C2×#106—28no—
136C2×#106C2318yes—
137C2×#106C2314yes—
138C2×#103—28no—
139C2×#103C2318yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#102—28no—
144C2×#102C2318yes—
145C2×#102C2314yes—
146C2×C22⋊C8D418yes—
147C2×C22⋊C8D418yes—
148C2×C22⋊C8C2312yes—
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2312yes—
151C2×C22⋊C8C2312yes—
152C2×C4×C8C2318yes—
153#400—24no—
154#400—28no—
155#400—28no—
156#397—24no—
157#394—24no—
158#393—24no—
159#391—24no—
160#391C2314yes—
161#391C2318yes—
162#390C2314yes—
163#388—28no—
164#386—28no—
165#386—24no—
166#386—24no—
167#382—24no—
168#382C2314yes—
169#382C2318yes—
170#381—24no—
171#381—28no—
172#381—28no—
173#379—24no—
174C4×C22⋊C4D412yes—
175C4×C22⋊C4D412yes—
176C4×C22⋊C4C2312yes—
177C4×C22⋊C4C2312yes—
178C2×#95—24no—
179C2×#95—24no—
180C2×#95D412yes—
181#336C2318yes—
182C25—21no—
183C25C2×D411yescharacteristic
184C22×Q8—21no—
185C22×Q8—21no—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8—22no—
190C22×Q8—44no—
191C22×Q8—44no—
192C22×Q8C2412yes—
193C22×Q8C2412yes—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—21no—
199C22×D4—44no—
200C22×D4—44no—
201C22×D4—48no—
202C22×D4C2412yes—
203C23×C4—21no—
204C23×C4—21no—
205C23×C4—21no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—21no—
210C23×C4—21no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4—48no—
214C23×C4C2×D414yes—
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D414yes—
219C23×C4C2×D414yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C2×Q16—28no—
224C2×Q16—48no—
225C2×SD16—28no—
226C2×SD16—48no—
227C22×C8—44no—
228C22×C8—44no—
229C22×C8C2×D418yes—
230C22×C8C2×D418yes—
231C22×C8C2414yes—
232C22×C8C2414yes—
233#124—48no—
234C22⋊Q8—24no—
235C22⋊Q8—24no—
236C22⋊Q8—24no—
237C22⋊Q8—24no—
238C22⋊Q8—24no—
239C22⋊Q8—24no—
240C22⋊Q8—24no—
241C22⋊Q8—24no—
242C22⋊Q8—44no—
243C22⋊Q8—44no—
244C22⋊Q8—44no—
245C22⋊Q8—44no—
246#121—24no—
247#121—24no—
248#121—24no—
249#121—24no—
250#121—44no—
251#121—44no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4C2414yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4C2×D414yes—
296C2×C22⋊C4C2×D414yes—
297C2×C42—44no—
298C2×C42C2×D414yes—
299C2×C42C2414yes—
300#107—216no—
301#107—28no—
302#106—216no—
303#106—28no—
304#103—216no—
305#103—28no—
306#103—28no—
307#103—28no—
308#102—216no—
309#102—28no—
310C22⋊C8C2×D418yes—
311C22⋊C8C2×D418yes—
312C22⋊C8C2×D418yes—
313C22⋊C8C2×D418yes—
314C4×C8—48no—
315#95—216no—
316#95—216no—
317#95—28no—
318#95—28no—
319#95C2×D414yes—
320#95C2×D414yes—
321C24—22no—
322C24—42no—
323C24—42no—
324C24—42no—
325C24—44no—
326C24—82no—
327C24—82no—
328C24#12714yes—
329C24#13518yes—
330C24C22×D411yescharacteristic
331C24C22×D411yescharacteristic
332C24C22×D411yescharacteristic
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—28no—
338C2×Q8—28no—
339C2×Q8—44no—
340C2×Q8—48no—
341C2×Q8—42no—
342C2×Q8—42no—
343C2×Q8—82no—
344C2×Q8—84no—
345C2×Q8—84no—
346C2×Q8—84no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—28no—
350C2×D4—48no—
351C2×D4—44no—
352C2×D4—44no—
353C2×D4—48no—
354C2×D4—48no—
355C2×D4—42no—
356C2×D4—84no—
357C2×D4—82no—
358C2×D4—816no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—28no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—48no—
370C22×C4—48no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—48no—
387C22×C4—48no—
388C22×C4—48no—
389C22×C4—44no—
390C22×C4—82no—
391C22×C4—82no—
392C22×C4—816no—
393C22×C4#12714yes—
394C22×C4#12714yes—
395C22×C4#12714yes—
396C22×C4C22×D418yes—
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D412yes—
401C22×C4C22×D412yes—
402C22×C4C22×D412yes—
403C22×C4C22×D412yes—
404C22×C4C22×D412yes—
405C22×C4C22×D412yes—
406C22×C4C2511yescharacteristic
407Q16—88no—
408SD16—88no—
409C2×C8—28no—
410C2×C8—28no—
411C2×C8—48no—
412C2×C8—48no—
413C2×C8C22×D418yes—
414C2×C8C22×D418yes—
415C4⋊C4—22no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—44no—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—44no—
430C4⋊C4—44no—
431C4⋊C4—44no—
432C4⋊C4—44no—
433C22⋊C4—28no—
434C22⋊C4—28no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—88no—
442C22⋊C4—88no—
443C42—28no—
444C42—44no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—44no—
452C23—44no—
453C23—416no—
454C23—44no—
455C23—42no—
456C23—44no—
457C23—44no—
458C23—84no—
459C23—84no—
460C23—84no—
461C23—168no—
462C23#495116yes—
463C23C2×#12712yes—
464C23C2×#12712yes—
465C23C2×#12712yes—
466C23C2×#13514yes—
467C23C2×#13514yes—
468C23C2×#13514yes—
469C23C23×D411yescharacteristic
470Q8—44no—
471Q8—44no—
472Q8—44no—
473Q8—44no—
474Q8—84no—
475Q8—84no—
476D4—44no—
477D4—44no—
478D4—84no—
479D4—88no—
480D4—88no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—28no—
491C2×C4—28no—
492C2×C4—28no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—48no—
507C2×C4—44no—
508C2×C4—88no—
509C2×C4—84no—
510C2×C4—84no—
511C2×C4—84no—
512C2×C4—84no—
513C2×C4—88no—
514C2×C4—88no—
515C2×C4—88no—
516C2×C4—84no—
517C2×C4C2×#12712yes—
518C2×C4C2×#12712yes—
519C2×C4C2×#12712yes—
520C2×C4C23×D411yescharacteristic
521C2×C4C23×D411yescharacteristic
522C2×C4#58514yes—
523C8—28no—
524C8—28no—
525C22—28no—
526C22—28no—
527C22—28no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—416no—
535C22—88no—
536C22—82no—
537C22—82no—
538C22—82no—
539C22—168no—
540C22C2×#49514yes—
541C22C2×#49514yes—
542C22C2×#49514yes—
543C22C22×#12711yescharacteristic
544C22#348512yes—
545C22C22×#13512yes—
546C22#358612yes—
547C4—22no—
548C4—22no—
549C4—22no—
550C4—48no—
551C4—82no—
552C4—82no—
553C4—84no—
554C4—82no—
555C4—82no—
556C4—82no—
557C4#348512yes—
558C4#348512yes—
559C2—24no—
560C2—24no—
561C2—24no—
562C2—162no—
563C2#3546311yescharacteristic
564C2C22×#49511yescharacteristic
565C2#6064511yescharacteristic
566C2#6169614yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#3189C2semidirect
3. C22×#483C2semidirect
6. C2×#2503C2semidirect
7. #28335C2semidirect
8. #28330C2semidirect
9. #28306C2semidirect
10. #27931C2semidirect
11. #27926C2semidirect
12. #27912C2semidirect
13. #20523C2semidirect
20. C2×#484C22semidirect
21. C2×#484C22semidirect
26. C2×#483C22semidirect
32. C2×#451C22semidirect
33. C2×#451C22semidirect
37. C22×#106C22semidirect
38. C22×#103C22semidirect
51. #2503C22semidirect
56. C2×#391C22semidirect
57. C2×#382C22semidirect
60. #2097C22semidirect
61. #2097C22semidirect
62. #2092C22semidirect
63. #2015C22semidirect
64. #1969C22semidirect
65. #1968C22semidirect
66. #1958C22semidirect
67. #1846C22semidirect
69. C8×C22⋊C4C22semidirect
70. C4×C22⋊C8C22semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#31892
C2×#31872
C22×#4832
C22×#4802
#350192
C2×#25032
#283352
#283302
#283062
#279312
#279262
#279122
#205232